To investigate influence mechanism of the second viscosity on internal flow of a normal shock wave, one-dimensional Navier-Stokes equations are numerically solved. It indicates that the second viscosity has a smoothing effect on density, heat flow and energy distribution in the shock wave, which results in a decrease of peak value of heat and entropy flows, and an increase of shock thickness. Due to the production of normal viscous dissipation, some lost mechanical energy is converted into internal energy. As considering the second viscosity, density distribution and shock thickness are greatly improved. They are in good agreement with experimental data. In addition, Knudsen number is obtained 0.12≤Kn≤0.4 within Mach number range from 1.2 to 10. It indicates that Navier-Stokes equations with the second viscosity simulate normal shock structure more accurately.

With kinetic and continuum theories of bulk viscosity coefficient, bulk-viscosity effect on two-dimensional toroidal shock-wave focus (Mach number Ma=2.0) is studied numerically. It shows that bulk-viscosity effect on toroidal shock-wave focusing is not negligible for perfect gases. Due to bulk viscous effect in compressible flows, pressure, temperature and density at the central focus perform 20% reduction, 10% increase and 30% reduction, respectively. Compared with rotational mode, shock wave focusing present obvious bulk-viscosity effect in vibrational mode, since bulk viscous stress has same order of magnitude with thermodynamic pressure.

According to advection upstream splitting method,a flux splitting method called K-CUSP is proposed.The greatest difference between K-CUSP and two traditional CUSP schemes,namely H-CUSP and E-CUSP,is splitting of total energy:All kinematic quantities and thermodynamic quantities should be separately split into convective term and pressure term by K-CUSP scheme.Numerical tests indicate that:① K-CUSP scheme inherits the simplicity and robustness of FVS scheme.It is less prone to pressure overshoot after shock and no oscillations in expansion area,which is better than AUSM and WPS schemes.② K-CUSP scheme also inherits resolution of FDS scheme.Shock resolution is almost the same with H-CUSP and E-CUSP schemes.Contact discontinuity resolution is better than FVS schemes,a little worse than Roe,AUSM and WPS schemes.However,velocity of contact discontinuity in AUSM and WPS schemes exist large oscillation,while our scheme does not.